Balloon

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Gas.Volumes.Balloon

Description

The component represents a closed volume of variable size enclosed by a fictious skin. This extension is introduced to model the sharp pressure increase when the volume approaches the maximum volume without altering the model equations.

Mass Balance

$\begin{equation*} \frac{dn_{i}}{dt} = F_{i} \end{equation*}$

with initial conditions depending on the selected thermodynamic model

Ideal Gas

$\begin{equation*} n_{i}(t=0) = \frac{x_{i_{0}} \, p_{0} \, V_{0}}{R \, T_{0}} \quad \text{for} \quad i=1,\dots,N \end{equation*}$

Peng Robinson

$\begin{equation*} n_{i}(t=0) = \frac{x_{i_{0}} \, p_{0} \, V_{0}}{z(p_{0},T_{0}) \, R \, T_{0}} \quad \text{for} \quad i=1,\dots,N \end{equation*}$

Energy Balance

$\begin{equation*} \sum_{i}^{N} F_{i} \, {\overline H}_{i}(T) + \Delta H_{res} \, \sum_{i}^{N} F_{i} \, + \left( \sum_{i}^{N} n_{i} \, c_{p_{i}}(T) \right) \, \frac{dT}{dt} = \Phi + \dot{Q} + V \, \frac{dp}{dt} \end{equation*}$

with initial condition

$\begin{equation*} T(t=0) = T_{0} \end{equation*}$

Equation of State

Pressure

The pressure acting on this volume is either given by a component parameter or set by a physical input signal which may even be time variant (see Assumptions).

$\begin{equation*} p = \Big( \left\{ \begin{array}{lcl} p_{0} \quad \text{or} \\ p_{in} & & \end{array} \right. \Big) \, \exp\left\{\frac{k \, V}{V_{max}-V} \right\} \end{equation*}$

The factor $k$ is internally set to $1.0 \times 10^{-4}$ .

Volume

Ideal Gas

$\begin{equation*} V = \sum_{i}^{N} n_{i} \, \frac{R \, T}{p} \end{equation*}$
Peng Robinson

$\begin{equation*} V = z \, \sum_{i}^{N} n_{i} \, \frac{R \, T}{p} \end{equation*}$

Assumptions and Limitations

The time response of the pressure input signal is assumed to be much slower than the dynamics of the balance component. Thus the pressure is regarded as approximately constant.
In some cases considering the expression work explicitly may slow down the convergence rate.
If the pressure is read from the physical input port, it must hold: $p_{0} = p_{in}(t=0)$ .

Ports

Conserving

Gas conserving ports
```
Port_A = Gas;  %
```
Gas conserving ports
```
Port_B = Gas;  %
```
The port is only visible when the option enable2Port is set to On.
Thermal conserving port
```
Port_C = foundation.thermal.thermal;  %
```
Dependencies: The port is only visible when isothermalOperation is set to Off.

Input

Physical signals that controls the pressure.
```
pin = {0,'bar'}; % p
```
Dependencies: The port is only visible when pressureInput is set to ON.

Output

Physical signal that represents the current volume
```
Vout = {0,'l'}; % V
```
Dependencies: The port is only visible when volumeInputOutput is set to Output.

Parameters

Options

Option to select pressure input
```
pressureInput = OnOff.Off; %
```
Off | On
Option to select Vmax input
```
VmaxInput = OnOff.Off; %
```
Off | On
Option to select volume output
```
volumeOutput = OnOff.Off; 
```
On Off
Option to select compression work
```
compressionWork = OnOff.On;      
```
Off | On

If the option is set to Off the term $V \, \frac{dp}{dt}$ in the energy balance will be skipped.
Option to select thermal behaviour of the volume.
```
isothermalOperation = OnOff.On;  
```
Off | On
Option to enable 2nd Port
```
enable2ndPort = OnOff.Off;    
```

Geometry

Initial volume
```
V0 = {1,'l'};       % Volume
```
Maximum Volume
```
Vmax0 = {1,'l'};   % Maximum Volume
```
The parameter is onyl visible when the option VmaxInput is set to Off.

Operating Conditions

Initial mole fractions
```
x0 = {[0;1],'1'};  
```
Note Initially only two species are considered. As the number of species can be changed via the properties dialogue, the size of the array must be adjusted accordingly.
Initial pressure
```
p0 = {1.0,'bar'};   % Initial Pressure
```

Initial Temperature

T0 = {298.15,'K'};  % Initial Temperature

Nominal Values

Nominal Values for Number of Moles
```
n_nom = {1,'mol'};
```
Nominal Value for Volume
```
V_nom = {1,'l'};
```

Nomenclature

$a,b$	EoS parameters
$c_{p_{i}}$	specific heat for species A_i
$F_{i}$	molar flow rate of species A_i
${\overline H}_{i}(T)$	molar enthalpy of species A_i
$\Delta H_{res}$	departure enthalpy of the mixture
$N$	total number of species
$n_{i}$	number of moles of species A_i
$p$	pressure
$p_{0}$	initial pressure
$Q$	heat flow rate (independent of fluid flow)
$R$	universal gas constant
$t$	time
$T$	temperature
$T_{0}$	initial temperature
$V_{0}$	volume
${\overline V}$	molar volume of mixture
$x_{i}$	mole fraction of species A_i
$x_{i_{0}}$	initial mole fraction of species A_i
$z$	compressibility
$\Phi$	energy flow rate